Factor the following expression: $x^2 - 9x + 8$
Solution: When we factor a polynomial, we are basically reversing this process of multiplying linear expressions together: $ \begin{eqnarray} (x + a)(x + b) &=& xx &+& xb + ax &+& ab \\ \\ &=& x^2 &+& {(a + b)}x &+& {ab} \end{eqnarray} $ $ \begin{eqnarray} \hphantom{(x + a)(x + b) }&\hphantom{=}&\hphantom{ xx }&\hphantom{+}&\hphantom{ (a + b)x }&\hphantom{+}& \\ &=& x^2 & & {-9}x& +& {8} \end{eqnarray} $ The coefficient on the $x$ term is $-9$ and the constant term is $8$ , so to reverse the steps above, we need to find two numbers that add up to $-9$ and multiply to $8$ You can try out different factors of $8$ to see if you can find two that satisfy both conditions. If you're stuck and can't think of any, you can also rewrite the conditions as a system of equations and try solving for $a$ and $b$ $ {a} + {b} = {-9}$ $ {a} \times {b} = {8}$ The two numbers $-1$ and $-8$ satisfy both conditions: $ {-1} + {-8} = {-9} $ $ {-1} \times {-8} = {8} $ So we can factor the expression as: $(x {-1})(x {-8})$